0. Tutorial 2: Game Theory
Matthew Robson

1. Q 2.1 Nash Equilibrium Ben 𝑏 1 𝑏 2 Ann 𝑎 1 2 , 5 6 , 2 𝑎 2 5 , 0 4 , 7
Here is a two-player game in which players Ann and Ben each choose, simultaneously, one of two strategies. The payoffs shown are vNM utilities.
Ben
𝑏 1
𝑏 2
Ann
𝑎 1
2 , 5
6 , 2
𝑎 2
5 , 0
4 , 7 1

2. Q 2.1 Nash Equilibrium Ben 𝑏 1 𝑏 2 Ann 𝑎 1 2 , 5 6 , 2 𝑎 2 5 , 0 4 , 7
How many pure-strategy Nash equilibria are there?
A: None
B: One
C: Two
D: Three
E: Four
Nash Equilibrium: Each player’s strategy is a best response to the strategy of the other player.
Ben
𝑏 1
𝑏 2
Ann
𝑎 1
2 , 5
6 , 2
𝑎 2
5 , 0
4 , 7 2

3. Q 2.1 Nash Equilibrium Ben 𝑏 1 𝑏 2 Ann 𝑎 1 2 , 5 6 , 2 𝑎 2 5 , 0 4 , 7
(b) The game has a mixed-strategy Nash equilibrium, where Ann chooses 𝑎 1 with probability α, and Ben chooses 𝑏 1 with probability β. Identify it:
A: α= 0.5 β= 0.5
B: α= 0.2 β= 0.5
C: α= 0.5 β= 0.2
D: α= 0.4 β= 0.7
E: α= 0.7 β= 0.4
Ben
𝑏 1
𝑏 2
Ann
𝑎 1
2 , 5
6 , 2
𝑎 2
5 , 0
4 , 7 3

4. Q 2.1 Nash Equilibrium
A mixed-strategy assigns a probability to each pure strategy, α and β. Allowing the player to randomly select a pure strategy.
What happens if we look at (a), what would Ann choose?
Randomising is rational for A iff she is different between 𝑎 1 and 𝑎 2 .
2𝛽+6 1−𝛽 =5𝛽+4 1−𝛽  𝛽=0.4
Randomising is rational for A iff she is different between 𝑏 1 and 𝑏 2 .
5𝛼+0 1−𝛼 =2𝛼+7 1−𝛼  𝛼 =0.7
Therefore:
E: α= 0.7 β= 0.4 4

5. Q 2.2 Prisoners Dilemma Consider this scenario:
The payoff matrix shown is based on the assumption that A and B each prefer fewer years in prison for themselves, not caring at all about the other. Given these payoffs, the dominant strategy for each player is to confess, even though they would both prefer that they both say nothing. This payoff structure defines the Prisoner’s Dilemma. B
Confess
Say Nothing A
A: 5 years in prison
B: 5 years in prison
A: goes free
B: 8 years in prison
A: 8 years in prison
B: goes free
A: 1 years in prison
B: 1 years in prison 5

6. Q 2.2 Prisoners Dilemma
Represent the utility preferences in this form:
Where: 8 years in prison ≺ 5 years in prison ≺ 1year in prison ≺ goes free
Is now: ≺ ≺ ≺ B
Confess
Say Nothing A
1 , 1
3 , 0
0 , 3
2 , 2 6

7. Q 2.2 Prisoners Dilemma
Here are two alternative assumptions about preferences:
(a) A and B each prefer fewer years in prison for the other, not caring at all about themselves.
(b) A and B each care only about the combined number of years they spend in prison (e.g. 10 years in the event that they both confess), preferring less to more.
In each case:
(i) construct the payoff matrix corresponding to the assumed preferences;
(ii) identify all the pure-strategy Nash Equilibria in the game;
(iii) explain why the game is not (despite the otherwise identical scenario) a Prisoner’s Dilemma;
(iv) consider whether there is any simple (and plausible) adjustment that could be made to the scenario, and in particular to the outcome in the event that one confesses and the other does not, that could induce A and B both to confess. 7

8. Q 2.2 Prisoners Dilemma
(i) (ii) Represent the utility preferences in this form:
iii) This resembles a PD in that each player has a dominant strategy. But in a PD there is an outcome preferred by both players to the equilibrium outcome, which is not the case here.
(iv) One simple possibility is: if only one confesses then that person will get 8 years, and the other allowed to go free. Given the preferences assumed here, this game is a PD, as shown before B
Confess
Say Nothing A
1 , 1
0 , 3
3 , 0
2 , 2 8

9. Q 2.2 Prisoners Dilemma
(b) (i) (ii) Represent the utility preferences in this form:
(iii) Same answer as (a). Note that in this case the two players have exactly the same preference orderings of the four outcomes.
(iv) One simple possibility is: if only one confesses then that person will get (say) 4 years and the other will get 8 (or vice versa), similar to a stag hunt, shown on the next slide. B
Confess
Say Nothing A
0 , 0
1 , 1
2 , 2 9

10. Q 2.2 Prisoners Dilemma B A Similar form to a Stag Hunt Game ConfessB
Confess
Say Nothing A
1 , 1
0 , 0
2 , 2 10

11. Q 2.3 Two Firms
Two firms share a market with a price (inverse demand) function:
p = ( 𝑞 𝑞 2 )
Each firm i (i = 1,2) has to choose a non-negative output 𝑞 𝑖 .
It has constant average cost 𝑐 𝑖 , its profit therefore being
𝜋 𝑖 = ( p- 𝑐 𝑖 ) 𝑞 𝑖 .
The market demand function and each firm’s cost are common knowledge to the two firms.
Assume that 𝑐 1 =2 and 𝑐 2 =4 . 11

12. Q 2.3 Two Firms
Using the price function, and the assumed 𝑐 𝑖 value, write each firm’s profit 𝜋 1 as a function of 𝑞 1 and 𝑞 2 .
For each firm, differentiate its profit function partially with respect to its own output 𝑞 𝑖 , set the partial derivative to zero, and then re- write the resulting equation to give that firm’s output as a function of the other firm’s output, i.e. respectively 𝑞 1 =𝑓( 𝑞 2 ) and
𝑞 2 =𝑓( 𝑞 1 ).
Sketch those two functions together in a diagram, with 𝑞 1 on the horizontal axis and 𝑞 2 on the vertical. 12

13. Q 2.3 Two Firms
𝑝=24 −0.2 𝑞 1 + 𝑞 2 , 𝜋 𝑖 = 𝑝− 𝑐 𝑖 𝑞 𝑖 , 𝑐 1 =2 , 𝑐 2 =4
𝜋 1 =(24 −0.2 𝑞 1 + 𝑞 2 − 𝑐 1 ) 𝑞 1 =22 𝑞 1 −0.2 𝑞 1 2 −0.2 𝑞 1 𝑞 2
𝜋 2 =(24 −0.2 𝑞 1 + 𝑞 2 − 𝑐 2 ) 𝑞 2 =20 𝑞 2 −0.2 𝑞 2 2 −0.2 𝑞 1 𝑞 2
𝜕 𝜋 1 𝜕 𝑞 1 =22−0.4 𝑞 1 −0.2 𝑞 2 =0
𝜕 𝜋 2 𝜕 𝑞 2 =20−0.2 𝑞 1 −0.4 𝑞 2 =0
𝑞 1 = 𝑞 =55−0.5 𝑞 2
𝑞 2 = 𝑞 =50−0.5 𝑞 1 13

14. Q 2.3 Two Firms
(b) Solve those two functions simultaneously, to give numerical solutions for 𝑞 1 and 𝑞 2 .
Locate the solution point in your diagram. Explain why this is a Nash Equilibrium.
Calculate each firm’s profit, and also the market price, at that equilibrium point. 14

15. Q 2.3 Two Firms 𝑞 1 +0.5 𝑞 2 =55 (1) 2𝑞 2 + 𝑞 1 =100 (2)
𝑞 𝑞 2 = (1)
2𝑞 2 + 𝑞 1 = (2)
2 − 𝑞 2 =45 → 𝑞 2 =30
𝑞 1 +15=55 → 𝑞 1 =40
𝑝=24 − =10
𝜋 1 = 10−2 40=320
𝜋 2 = 10−4 30=180 15

16. Q 2.3 Two Firms
(c) Assume that, before the firms choose their outputs, Firm 1 can offer to buy Firm 2.
If Firm 1 acquired Firm 2, why would it then choose 𝑞 2 =0?
What 𝒒 𝟏 would it choose, and what would be its profit?
Given this, what do your answers to (b) suggest is the maximum that Firm 1 would be willing to pay to acquire Firm 2, and the minimum that the current owners of Firm 2 would accept? 16

17. Q 2.3 Two Firms
If 𝑞 2 > 0 then a cost saving could be made, with no change to total output or revenue, by switching production from Firm 2 to Firm 1. So profit-maximisation requires 𝑞 2 = 0 . Given this, the profit- maximisation value of 𝑞 1 follows from (a): 𝑞 1 =55, p=13 , 𝜋 1 =605
It follows that the owners of Firm 1 would be willing to pay up to =285 to acquire Firm 2. And Firm 2’s owners would accept 180 or more. 17

18. Q 2.3 Two Firms
(d) Assume instead that Firm 1 chooses and reveals its output 𝑞 1 , before Firm 2 chooses its output 𝑞 2 , and that each firm has only two possible levels of output from which to choose:
Firm 1: 𝑞 1 =35 or 𝑞 1 =50
Firm 2: 𝑞 2 =15 or 𝑞 2 =20
Calculate the market price, and each firm’s profit, for each of the four possible combinations of 𝑞 1 and 𝑞 2 . Using these values as payoffs, construct an extensive form game tree. Then solve the game by backward induction.
Remember: 𝑝=24 −0.2 𝑞 1 + 𝑞 2 , 𝜋 𝑖 = 𝑝− 𝑐 𝑖 𝑞 𝑖
𝑐 1 =2 and 𝑐 2 =4 . 18

19. Q 2.3 Two Firms Backward induction:
𝑞 1 =50, 𝑞 2 =20, 𝑝=10, 𝜋 1 =400, 𝜋 2 =120 19

20. Q 2.3 Two Firms
(d)
Explain why both firms would benefit if Firm 2 was able to make a binding commitment to a strategy before Firm 1 made its output choice, and to make this commitment known to Firm 1.
Exactly what strategy would Firm 2 commit to? Would the consumers in this market also benefit from Firm 2 being able to make such a commitment? 20

21. Q 2.3 Two Firms If possible, Firm 2 commits to strategy of:
if 35 then 15; if 50 then 20
If Firm 1 can rely on this commitment, then
𝑞 1 =35, 𝑞 2 =15, 𝑝=14, 𝜋 1 =420, 𝜋 2 =150
But note that an unconditional commitment to 15 would instead induce 𝑞 1 =50. Consumers are worse off with the latter, because higher price and lower quantity. 21